Final Round of National Olympiad

It suffices to show that the infinite decimal fraction is periodic. For that, note that each new digit except for the first digit is congruent to twice the previous digit modulo $10$. Indeed, let $a_1,\dots ,a_{k-1},a_k$ be the existing at some time moment digits where $a_k$ is already added by Juku. Then the next digit $a_{k+1}$ satisfies $$a_{k+1}\equiv a_1+\cdots +a_k=(a_1+\cdots +a_{k-1})+a_k\equiv a_k+a_k=2a_k(\mathrm{mod}10).$$ Hence each new digit is uniquely determined by the last existing digit. As there are only a finite number of different digits, some digit must be added repeatedly. According to the fact just proven, all following digits are repeated as well.